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polymer nanocomposites: The active role of the matrix in stiffening mechanics
Accepted Manuscript Graphene/polymer nanocomposites: the active role of the matrix in stiffening mechanics Abdelrahman Hussein, Byungki Kim PII: DOI: Reference:
S0263-8223(17)33319-6 https://doi.org/10.1016/j.compstruct.2018.01.023 COST 9263
To appear in:
Composite Structures
Received Date: Revised Date: Accepted Date:
11 October 2017 23 December 2017 9 January 2018
Please cite this article as: Hussein, A., Kim, B., Graphene/polymer nanocomposites: the active role of the matrix in stiffening mechanics, Composite Structures (2018), doi: https://doi.org/10.1016/j.compstruct.2018.01.023
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Graphene/polymer nanocomposites: the active role of the matrix in stiffening mechanics Abdelrahman Hussein ([email protected]) and Byungki Kim* ([email protected]) School of Mechatronics Engineering, Korea University of Technology and Education, 1600 Chungjeol-ro, Byeongcheon-myeon, Dongnam-gu, Cheonan, Chungnam, 31253 Republic of Korea. *
Corresponding author
Abstract In this study we investigate the efficiency of graphene in stiffening polymer matrix nanocomposites. The analysis focuses primarily on the effect of the matrix modulus on the strain fields, whilst assuming perfect graphene and interface conditions with no agglomeration. The matrix was found to have an active role in stiffening due to the regions with large strain levels. The upper and lower bounds were discussed from the perspective of strain distribution. The effective modulus of graphene nanocomposite was relatively closer to the lower bound, while it showed a transitional behavior towards the upper bound with increasing and the volume fraction. The matrix contribution to stiffening surpasses the graphene in terms of the strain energy due to the graphene’s low strain levels resulting from the large modulus-mismatch. A novel measure to the internal state of strain that could represent the strain variation between the graphene and the matrix was derived. Lower modulus-mismatch show lower strain variation and a better stiffening efficiency of graphene. These measures could also interpret the effect of interphase on stiffening. This study provides new insights to the analysis and design of graphene/polymer nanocomposites. Keywords: Graphene; RVE-FEM; Mori-Tanaka; Strain-energy
1
Graphene is the strongest synthetic material with tensile modulus in the order of 1 TPa and tensile strength of 130 GPa [1]. Research efforts were directed towards utilizing these impressive mechanical properties in nanocomposites [2–6]. Yet, it is reported that the reinforcement levels of nanoparticles like graphene and carbon nanotubes are “generally disappointing” [7] and might lead to the loss of structural integrity resulting in lower damage tolerance [8]. Young et al. [7] surveyed the stiffening , defined as the ratio of uniaxial composite modulus to that of the matrix / , and found that rigid matrices with ≈ 1 GPa showed ≈ 2, while more compliant ones with ≈1E-3 GPa showed ≈ 10. They discussed these results using the upper (Voigt) and lower (Reuss) bounds and attributed these results to imperfections in graphene platelets like incomplete exfoliation, agglomeration, weak interface and defects. In their very recent review [9], Young et al updated their survey and concluded that the common stiffening measure can be misleading. They deduced a direct relation between and the effective modulus of graphene, obtained by the rule of mixtures. In other words, they found that; opposite to what is common in literature that the stiffening efficiency of graphene is higher for compliant matrices, stiffer matrices show better stiffening. This is due to the utilization of the misleading measure . However, they did not provide a quantitative interpretation to this behavior from the perspective of the stiffening mechanics. Modelling and simulation is frequently used to study nanocomposites due to the sophisticated nature of deformation measurements at the nanoscale [10]. Various atomistic and multiscale models were employed in the analysis of nanocomposites [11–13]. Although free standing graphene is flexible in the out-of-plane deformation [14], the flexural rigidity of graphene embedded in the polymer matrix increases 6 orders of magnitude [15]. Further, the deformation of graphene embedded in polymer matrix is linear under relatively low tensile and compressive loading [16–18]. The reinforcing ability of graphene in polymer 2
nanocomposites was evidenced by the interfacial load-transfer following the shear-lag mechanism [19]. Thus, continuum based micromechanics framework is valid for modeling the load-transfer and the mechanics of nanocomposites [5,19–21]. In this framework, the effects of several parameters including: volume fraction, orientation, interface, aspect ratio and agglomeration were investigated [22–25]. In these studies, the effect of the matrix modulus is only discussed in terms of without considering the analysis of the microscopic strain fields assuming that the small size of the nano reinforcements will not affect the matrix [26,27]. Additionally, the matrix can interact with nanofillers at the interface resulting in confined interfacial regions or inducing transcrystallinity for semi-crystalline polymers [28–31]. These regions are called the interphase and have different thermomechanical properties than the bulk matrix [32]. It was reported that these interphase regions improve the glass transition temperature [32], modulus [33], and fracture toughness [34]. Mortazavi et al. [35] showed that the interphase has a strong stiffening effect for spherical nanoparticles, while for high aspect ratio particles like graphene and carbon nanotubes the stiffening effect is insignificant. This is in agreement with investigations on clay nanoparticles using Finite Element Modeling (FEM) [36]. However, to the best of the authors’ knowledge, the effect of interphase surrounding graphene particles on load-transfer and stiffening of polymer nanocomposites is lacking in the literature. This paper is motivated by two questions. First, what are the maximum stiffening levels that could be achieved by utilizing graphene as a reinforcement? And second, what is the role of the matrix in stiffening nanocomposites, and can it have an active contribution? We address these questions by analyzing the strain fields calculated from micromechanics based FEM. A simple formula is derived to express the strain levels within a general composite based on strain-energy. Unlike the average strain theorem, this formula could express the 3
effect of elastic properties of the constituent materials, the volume fraction of the reinforcement and the applied strain at the boundaries. The active role of the matrix is investigated in a single particle composite model then extended to 3D randomly oriented composite. The results were compared to the Voigt and Reuss bounds to investigate the efficiency of graphene in stiffening. Additionally, the effect of interphase thickness and modulus was investigated.
1. Background Indicial notation is mainly used in this article. The superscripts m, f and i are used for the matrix, reinforcement and interphase respectively. The case of no superscripts indicates the composite phase, while the case of no subscripts indicates the property along the loading 11 direction. Matrix notation is used whenever necessary. Upper-case bold-face Roman letters are used exclusively for fourth-order tensors. For composite materials, calculations were performed on a statistically representative domain of the composite microstructure called the representative volume element (RVE) with volume . The microscopic elastic-fields are non-uniform due to the perturbations of the reinforcement phase. Effective macroscopic elastic-fields are calculated through the volume averaging of the micro-fields as: =
1
(1)
Where is the volume averaged microscopic field . Unfortunately, the analytical solution of the micro-fields is a formidable task [37], therefore FEM is used for these calculations. For such a discrete system, volume averaging is calculated from: =
1
4
(2)
Where and are the field and volume of the element respectiveley. Following the direct approach, the effective stiffness tensor of the composite can be expressed as [37]: =
(3)
Where and are the volume-averaged stress and strain tensors respectively. Finding can be interpreted as finding an energetically equivalent homogenous material with strain-energy such that: 1 1 1 = = = 2 2 2
(4)
The effective stiffness tensor of the composite could be also approximated by Meanfield homogenization (MFH) methods based on semi-analytical models like the Mori-Tanaka (MT) model. However, MFH methods do not calculate the micro-fields of the RVE. For an isotropic matrix stiffness tensor reinforced by a filler with stiffness tensor
, the effective elastic properties are bounded by Hill’s bounds that could be represented in terms of the bulk and shear moduli as [38]: ≤ ≤ + ( − ) − ( − ) ≤ ≤ + ( − ) − ( − )
(5)
Where is the filler volume fraction. Nevertheless, in our analysis we will use the more common - yet coarse - approximation of the bounds in terms of Young’s modulus , since this form is physically appealing in describing the deformation mechanics associated with the bounds. The bounds can be thought of a representation of the arrangement or the configuration of the reinforcement within the matrix, that influences the distribution of the strain fields as illustrated schematically in Figure 1. The Voigt bound represents the state of
5
uniform strain where the strain in the composite is equal to that in the matrix and the reinforcement, such that = = (Figure 1a). The Voigt bound could be modeled by the rule-of-mixtures: = + (1 − )
(6)
The Reuss lower bound represents a state of uniform stress, such that = = (Figure 1c). The Reuss bound could be expressed as: 1 − $% =" + #
(7)
Nano reinforcements like graphene are arranged in short-fiber or particle configuration as shown in Figure 1b. The strain field in this configuration is non-uniform, which is the reason for the difficulty in analytical calculations. Therefore, in this study will be calculated from FEM using the direct approach Eq. (2) and (3) as well as the MT model as will be discussed in section 3.
Figure 1: Schematic representation of the main three composite configurations and their corresponding strain fields in the vicinity of the reinforcement.
2. Average strain squared (AVSS) as a measure of internal strain levels By carefully observing the particle configuration in Figure 1b, increasing the modulusmismatch between graphene and the matrix will result in reduced strain levels in the 6
graphene. As a result, the matrix regions at the edges of the graphene platelet will be overstrained to compensate for this reduction. When only macroscopic fields are considered, the average-strain theorem [38] applies, which states as: let the RVE domain be subjected to a constant far-field strain ∘ prescribed on its boundary ∂, then the volume average of the strain is constant and is equal to ∘ : = ∘
(8)
Thus, it couldn’t express the effect of the composite parameters on the internal strain levels. From the equivalence of the direct and the energy approaches (see supplementary information S1), the strain levels within a general composite could be expressed in terms of the average-strain-squared ( (AVSS) that could be approximated by: ( ( + (1 − ) = (
(9)
Eq. (9) shows that the internal strain level could be expressed in terms of AVSS, which is a function of ∘ , f and more important, the elastic properties. The lower the , the larger the strain within the composite. Furthermore, AVSS is always larger than the square of the (
average-strain except for the Voigt configuration such that: ( ≥
(
(10)
The average-strain theorem is evaluated at the boundary ∂ of the RVE through Gauss theorem. Therefore, it represents only the strain state at the boundaries, which is essentially constant. On the other side, the variation of strain between the matrix and the reinforcement is a local effect within the RVE domain. The advantages of AVSS over the average-strain theorem is that it shows the effect of different composite material parameters on the internal strain levels. As will be shown later in this article, this could be used as a measure of the efficiency of the reinforcement in stiffening. 7
3. The RVE model 3.1 The FEM model The RVE was modeled in Digimat-FE v2016.1 (Extreme Engineering, MSC Software, Belgium). The graphene was modeled as a disc [23,24,39] with an aspect ratio of ≈500. For the single particle model, the RVE is shown in Figure 2 with f of 0.387%. The FEM model was solved in ABAQUS v6.14 (Simulia, Dassault Systèmes, France). The RVE was meshed with tetragonal C3D10M elements. Since we are attempting the theoretically maximum limits of the particle configuration, the graphene was perfectly bonded to the matrix. All the models were subjected to 1.0E-3 uniaxial strain in the 11 direction. The material properties for graphene were 1.0 TPa for Young’s modulus and 0.165 for Poisson’s ratio [1]. The matrix moduli were varied between 0.25 and 2.5 GPa with Poisson’s ratio of 0.3. Additionally, we investigate the effect of aspect ratio of graphene platelet between 100 and 500 at constant f. In the FEM model, the interphase was modeled as a coating to the graphene platelets with a constant thickness. We investigated the effect of changing the interphase modulus values between 3.0 and 5.0 GPa as well as the thickness * of up to three times that of the graphene platelet, i.e. *, 2* and 3*.
Figure 2: Single-particle RVE with a cut-through showing the graphene platelet. For the 3D randomly oriented particles, we used a cubic RVE as shown in Figure 3. For this model, we used the same parameters except for f values that were in the range of 0.640.05 vol%.
8
Figure 3: Representative 3D randomly oriented particles RVE with f=0.64 % left and right the graphene distribution within. 3.2 Mean-field homogenization For the MFH calculations, the MT stiffness tensor is expressed as [38]: ,-./ = , + (, − , )0./
(11)
Where 0./ is the MT concentration tensor: 0./ = 01 [(1 − )3 + 01 ]
%$(12)
Where I is the unit tensor and 01 is the dilute concentration tensor, which could be expressed in terms of the Eshelby tensor for disc-shaped inclusion S [40] as: 01 = [3 + 5[, ]$%(, − , )]
%$(13)
The details of calculating S could be found in the supplementary material S2. For modeling the interphase we used the two-level MT homogenization scheme [41]. As shown schematically in Figure 4, first at the deepest level, the graphene (inclusion 1) is homogenized with the interphase resulting in inclusion 2, which is then homogenized with the matrix resulting in the overall composite. The deepest-level graphene/interphase homogenization to obtain the stiffness matrix ,- 6 can be calculated from: ,-6 = , +
(, − , )78 +
9
(14)
Where is the interphase volume fraction and 78 is the graphene/interphase starin concentration tensor that could be expressed as: $% 78 = ( + ) 71 6 [ 3 + 5]
(15)
And 71 6 is:
%$ $% 71 6 = [3 + 59, : (, − , )]
(16)
The highest-level homogenization is performed through: ,-./ = , + ( + )(,-6 − , )78(
(17)
With 78( the inclusion 2/matrix concentration tensor: 1 $% 78( = 71 ( [ 3 + ( + )7( ]
(18)
$% -6 $% 71 ( = [3 + 5[, ] (, − , )]
(19)
And 71 ( :
Figure 4: Illustration of the two-level MT homogenization scheme to account for the interphase as a coating to the graphene For a 3D randomly oriented RVE, the MT stiffness tensor is expressed as [42,43]: ,-./ = (1 − ), 06 + {, 0./ }
(20)
Where the braces {} stand for orientation averaging of a 4 th order tensor of 3D randomly orientated particles.
10
4. Results of single particle model 4.1 Effect of matrix modulus and volume fraction The results of for the single particle RVE are shown in Figure 5. The MT model slightly underestimates the FEM results. MT showed ≈ 1.1, while the FEM results show ≈ 1.3. It was reported that for aligned particles, MT underestimates FEM, while for 3D randomly oriented particles MT overestimates FEM [39]. That will be shown to be the case for 3D randomly oriented particles in subsequent sections. Thus, the FEM predictions are in good agreement with the MT model and the micro-fields calculated from FEM could be used for further analysis.
Figure 5: values calculated for single-particle RVE from FEM and MT model. (
The relation between and the average-strain theorem in terms of as well as AVSS is shown in Figure 6. Both values were calculated from FEM according to Eq. (2). These results (
were compared to the LHS of the approximated AVSS in Eq. (9). The results show that is constant and independent of as predicted from the average-strain theorem. On the other side, ( is not constant and is inversely proportional to as predicted from AVSS. Eq. (9) represents a good approximation of AVSS. Further, the results are also in agreement with Eq. (
(10) since ( > .
11
Figure 6: The effect of on the values of strain-squared calculated from approximate ( AVSS in Eq. (9), ( and . The main outcome of AVSS is to highlight the effect of the on the strain levels within the composite. Next, we show how this would affect the stiffening mechanics. Due to the non-uniform strain fields, the contribution of each of the graphene and the matrix to could be calculated from their respective contribution to . In the case of isotropic material, the constitutive relation between stress and strain is written as: =
? ( + @ ) 1+? 1 − 2?
(21)
Thus, Eq. (4) could be written as: 1 ? = ( + ) 2 1+? 1 − 2?
(22)
Knowing that the engineering shear strain ABC = 2BC , Eq. (22) in component form becomes [44]: 1 1 ( ? ( ( = ((B ( +C ( + D ( + (ABC + ABD + ACD )) + ( + + D )( ) 2 1+? 2 1 − 2? B C
12
(23)
Eq. (23) can be applied to FEM results in the discrete form of Eq. (2). The contribution of the matrix to could be calculated from its fraction strain energy E . Given the volume of graphene platelet F, the strain energy in the matrix can be calculated by replacing V with − F for the integration domain. Then, the fraction strain energy in the matrix E could be defined as: E =
(24)
Similarly, the fraction-strain-energy in the graphene E is defined as: E =
(25)
The relation between the fraction strain energy and in Figure 7 shows E > 0.95 and E J 0.05. Unexpectedly, E is inversely proportional to , while E is directly proportional to . In fact, the contribution of graphene can be neglected at lower values of . Therefore, the fraction strain energy and results suggest the need for a detailed analysis of the strain fields for the particle configuration.
Figure 7: The effect of on E and E . The strain field in the loading direction is shown in Figure 8 for the matrix moduli = 0.25 and 2.5 GPa. The strain field distribution is very similar in both cases; however,
13
the magnitudes are different within the graphene and the matrix regions surrounding it as can be seen in Figure 8a and b. The values of strain in the graphene platelet and the matrix regions adjacent to it are significantly lower than those regions at the platelet edges. The strain field within the graphene is shown in Figure 8c and d for = 0.25 and 2.5 GPa respectively. The magnitude of strain increases from the platelet edges to a maximum plateau at the center following the shear-lag theory [19,45]. The plateau of the strain in graphene increases with increasing [36]. According to the shear-lag theory, the load is transferred to the graphene particle through shear forces at the interface with the matrix. When decreases, the shear stresses, and thus the shear forces decrease resulting in lower pull and lower strain levels in the graphene. Figure 8e shows the volume-averaged strain in the graphene as a function of . =1.0E-3 is constant and is shown for comparison. is directly proportional to and increases from 2.44E-5 to 1.99E-4 for = 0.25 and 2.5 GPa respectively, which is orders of magnitude lower than . This explains the low E of graphene despite its very high elastic properties and its response to .
14
Figure 8: Strain fields in a section through the RVE for (a) = 0.25 GPa and (b) = 2.5 GPa, (c) and (d) the corresponding strain-fields distribution in the graphene, and (e) the and as a function of . Representative values of with respect to the position from the graphene platelet is shown in Figure 9. The regions at the edges of the graphene platelets have > , while the adjacent regions have J . Additionally, in the regions adjacent to the graphene platelet is directly proportional to , while in the edge regions is inversely proportional to . This suggests that in the edge regions is responsible for the increased AVSS with decreasing . Additionally, the varying values leads to varying stress levels as shown in Figure 9b and c. The edge regions have high stress levels, and thus, higher resistance to deformation. Therefore, contrary to what was reported in [26,27], the small size of nanoreinforcements does not prevent the stress concentration at the edge regions, and these regions are responsible for the active stiffening role of the matrix and its high E .
15
Figure 9: (a) Strain at representative positions of the matrix with respect to the graphene and their corresponding stress values at (b) = 0.25 GPa and (c) = 2.5 GPa. 4.2 Effect of aspect ratio The effect of the aspect ratio on is shown in Figure 10a. As the aspect ratio increase, increases due to the enhanced load-transfer as shown in Figure 10b. This is in agreement with the shear-lag model [26,45] and FEM reports [23,24,36]. The enhanced load-transfer could be evidenced from increasing E with the aspect ratio Figure 10c. However, E shows an inverse behavior. From the analysis of these results, it is interesting to note the similarity between the effect of the aspect ratio and the modulus-mismatch on stiffening mechanics by using the measures E and E , and the increased active role of the matrix at lower loadtransfer levels to graphene.
Figure 10: Effect of aspect ratio for a single particle RVE on (a) , (b) and (c) E and E . 16
4.3 Effect of interphase The effect of the interphase modulus and thickness t on for a single particle RVE is shown in Figure 11a. These results are compared to the case of no interphase, which is represented by = =2.5 GPa. Here we used fixed = 2.5 GPa since it showed the highest as discussed in Figure 5. Moreover, for the single particle RVE, the two-level MT homogenization underestimates the FEM results. The increase of with increasing and * is insignificant for the single particle RVE. This is due to the small filler volume fraction f [46] that results in a small interphase volume fraction = 0.794% and 2.42% for t and 3* respectively. This could be explained by analyzing and E shown in Figure 11b and d respectively. The interphase shows a maximum increase of ≈2% in at 3* and =5.0 GPa. This results in a modest increase in E , which explains the insignificant effect on . The interphase does not only affect the load-transfer to the graphene, but also affects the matrix contribution in terms of E . Figure 11c shows that, E has a value of 0.951 that reduces abruptly when introducing the interphase. In the presence of interphase, E decreases with increasing the interphase thickness. This is due to the increase of with increasing t. On the other side, it is interesting to see that E decreases, although modestly, with increasing . Further, the fraction strain energy of the interphase E (Figure 11e) increases with increasing t, while it decreases with increasing , i.e increasing the modulus mismatch with the bulk matrix. This behavior is similar to the behavior of a reinforcing phase as discussed earlier in Figure 7. Therefore, the interphase contributes to the stiffening by two mechanisms. First, it acts as a buffer zone for enhanced load-transfer to the graphene, which increases with increasing and t as can be seen by observing . Secondly, the interphase itself contributes to stiffening similar to a reinforcing phase as can be seen from E as well as the increased E with 17
increasing . This shows that the interpahse behaves as a reinforcing phase when its properties are higher than the bulk matrix, i.e. > . This discussion provides a foundation to the analysis of the effect of interpahse in the 3D RVE with multiple graphene platelets which shows more significant contribution due to the increased as will be shown in section 5.4.
Figure 11: Effect of interphase modulus and thickness t for a single particle RVE on (a) , (b) , (c) E , (d) E and E . For (a-d) the value = 2.5 GPa represents no interphase, i.e. same value to the matrix . The legend is shown in (a).
5. Results of the 3D randomly oriented particles model 5.1 Generation of 3D RVE First we point out the effect of aspect-ratio of graphene on the maximum agglomerationfree volume fraction f of 3D randomly oriented particles. Based on the seeding algorithm of Digimat-FE, the maximum possible agglomeration-free f was 0.64%. Higher volume fractions could be achived for uniformly oriented particles or in the presense of clustering
18
[39]. This suggests that in addition to physicochemical interactions, the geometry affects the agglomeration tendency of the nanoparticles where higher aspect-ratio leads to higher tendency to agglomeration. From an experimental point of view, this would affect the feasibility of using higher aspect-ratio particles. Most of the synthesis methods result in randomly oriented nanoparticles. High aspect-ratio particles will easily form clusters that would reduce their effective aspect-ratio and result in poor stiffening. Thus, there should be a compromise between the aspect-ratio and the clustering-free volume-fraction. Further, the generation of 3D randomly oriented particles would inevitebly induce partial anisotropy. By analyzing the tension applied to the three normal directions 11, 22 and 33 to obtain the three moduli E11, E22 and E33 respectively (supplementary material Figure S.3). We found a correlation between the anisotropy and the global error on orientation tensor indicator [47] (see supplementary material S.3). This indicator was found to have a direct relation on the size of the RVE and the volume fraction f. On the other side, increaseing the RVE size significantly increases the meshing difficulty and computational cost. In this study we used an RVE with edge size 3.5 times the diameter of the filler. This ensures levels of global error on orientation tensor of no more than 0.09 to ensure the highest possible isotropy. 5.2 Effect of matirx modulus and volume fraction The effect of on and is shown in Figure 12. The Voigt and Reuss bounds are plotted to show how the configuration affects the strain distribution, which in turn affects the stiffening mechanics. Figure 12a shows that the Voigt configuration has the highest since the strain in graphene is maximum as = . The Voigt configuration has the highest , which increases exponentially with decreasing as shown in Figure 12d. increases from ≈ 3 to 25 for =2.5 and 0.25 GPa respectively at f=0.64%. On the other side, the Reuss
19
configuration shows ≈ (Figure 12c). The large modulus-mismatch makes the graphene to behave like a rigid body with almost no strain, and thus, no contribution to stiffening. This can be seen from its values of ≈1 that are independent of f. The values for the particle configuration are shown in Figure 12b. The MT model slightly overestimates the FEM results for 3D randomly oriented particles [39]. The dependence of on f increases with increasing . This indicates a transition towards the Voigt behavior; i.e. increased strain levels within the filler, with increasing . As in the Voigt configuration, increases with decreasing and is dependent on f. The values of ≈2 for f=0.64%, which is in agreement with the experimental results reported in [7,48] for a modulus-mismatch of ≈1.0E3.
Figure 12: The effect of on and for different configurations: (a) and (d) Voigt, (b) and (e) particle, and (c) and (f) Reuss. Solid lines represent the MT results for the particle configuration. The legend is shown in (c). The AVSS of the RVE model with 3D randomly oriented graphene particles is shown in Figure 13. ( shows the same behavior discussed earlier in Figure 6, that is, ( is inversely (
proportional to and that ( > . However, the gap with the approximate expression of AVSS in Eq. (9) is larger. This is due to the increased effect of 3D strain state due to the 3D random orientation of graphene particles. Additionally, Figure 13 shows that ( is directly
20
proportional to f as predicted by AVSS. When f increases, the number of graphene particles increases and the number of highly strained edge regions of the matrix increases. This results in an overall increase in ( . Thus, for a given ∘ and f, higher AVSS indicates higher variation between the strain levels in the matrix and the reinforcement.
Figure 13: The effect of on AVSS calculated from Eq. (9) (solid lines), and ( and calculated from FEM for different f for the RVE 3D randomly oriented particles.
(
The fraction strain energy values for the three configurations are shown in Figure 14. E is complementary to E , i.e. E + E = 1, hence we limit the discussion to E for brevity. The values of E are shown in Figure 14d-f for completeness. For the three configurations, E is inversely proportional to f. E in the Voigt configuration (Figure 14a) is directly proportional to . This is expected since in this configuration the strain is uniform. The graphene in the Voigt configuration showed a pronounced contribution to stiffening since for only f=0.5%, E ≈ 0.03 and 0.3 for =0.25 and 2.5 respectively. On the other side, for the particle and the Reuss configurations, E is inversely proportional to . The Reuss configuration shows E ≈ 1 in Figure 14c which is in agreement with the values discussed in Figure 12c. Nevertheless, the results show its comparable behavior to the particle configuration with respect to . The particle
21
configuration in Figure 14b has E ≈ 0.99 for f=0.05% that decreases with increasing f. At f=0.64% E ≈ 0.95 and 0.83 for =0.25 and 2.5 GPa respectively. These results show the significance of considering the strain fields when analyzing the stiffening mechanics of particle configuration system like nanocomposites. Unlike the Voigt configuration, the contribution of graphene to stiffening in the particle configuration decreases with decreasing . The matrix actively contributes to stiffening due to its highly strained regions as can be seen from its surpassing E values. This active contribution increases with decreasing as the matrix is overstrained to compensate for the decreasing strain levels within the graphene. This can be also demonstrated from the AVSS results, where the increased strain levels with decreasing indicated increased variation in strain levels between the matrix and the graphene.
Figure 14: Fraction strain energy for Voigt (a) and (d), particle (b) and (e), and Reuss (c) and (f) configurations. The legend is shown in (a). 5.3 Effect of aspect ratio For 3D RVE, we investigated the effect of aspect ratio and interphase (section 5.4) for the filler volume fraction f 0.5% and 2.5 GPa. The results here are similar to those discussed
22
for single particle RVE in Figure 10. This further verifies the importance of the analysis of strain fields and the use of E and E to understand the stiffening mechanics of graphene polymer nanocomposites.
Figure 15: Effect of aspect ratio for 3D RVE on (a) , (b) and (c) E and E . 5.4 Effect of interphase For the 3D RVE, the effect of interphase is summarized in Figure 16. Here, the interpahse thickness *, 2* and 3* corresponds to interphase volume fraction fi 1.03%, 2.08% and 3.14% respectively. The effect of interphase on is shown in Figure 16a. The two-level MT homogenization scheme over estimates the FEM results, however, it shows satisfactory estimations considering the simplicity of calculation compared to FEM. It should also be noted that the effect of the interphase on in the 3D RVE is very sensitive to the global error in orientation tensor that should be minimal in order to obtain reliable results. Compared to the single particle RVE, here, the thickness shows more efficiency to enhance as can be shown from the sudden increase form * to 2*. This behavior can also be observed for , E and E KL Figure 16(b-d). However, for E , there is no abrupt increase with *. This indicates that in addition to the role of the interphase discussed in section 4.2, there is an additional mechanism for multi-particle RVE, that is, increased interaction of strain fields with
23
increasing *. The larger interaction of strain fields enhances the strain levels in the filler that results in improving the stiffness .
Figure 16: Summary of the results of 3D RVE with f=0.5% and interphase thickness t: (a) , (b) , (c) E , (d) E and (e) E . For (a-d) the value = 2.5 GPa represents no interphase, The legend is shown in (a).
6. Discussion Mean-field homogenization with Mori-Tanaka scheme is convenient for estimating the effect of microstructural parameters like modulus-mismatch, aspect-ratio and interphase on the effective elastic properties of graphene polymer nanocomposites. This is due to its generality and relative ease of calculation compared to FEM. However, to understand the stiffening and load-transfer mechanics, microscopic fields are required, and thus, FEM becomes indispensable. The motivation for this study was to understand the stiffening mechanism in graphene nanocomposites, the limits to this stiffening with respect to the upper and the lower bounds and the contribution of each of the graphene and the matrix. depends not only on the elastic properties of the graphene and the matrix, but also on their respective strain levels. This 24
dependence is appropriately expressed in terms of strain energy based measures. The configuration of the graphene affects the distribution of the strain field and subsequently affects the bounds of . The Voigt bound is the upper bound because the strain in the reinforcement is maximum and is equal to that of the matrix, whilst it is minimum for the Ruess lower bound. The particle configuration, which represents most of the experimentally prepared graphene nanocomposites, shows values close to the lower bound. This is due to the graphene’s very low E as a result of its low . The E values further decrease with increasing the modulus-mismatch with the matrix. Moreover, due to its very large aspect ratio, the maximum agglomeration free volume fraction of 3D randomly oriented particles is very limited. It appears that the main two impressive characteristics of graphene, i.e. its modulus and aspect-ratio, hinder its efficiency in stiffening nanocomposites. For the Voigt configuration, matrices with low modulus show impressive values when reinforced with graphene due to its high mechanical properties. However, in the case of nonuniform strain fields like the particle configuration is a misleading measure [20]. The matrix is overstrained to compensate for the lower due to the large modulus-mismatch. Thus, the matrix plays an active role in stiffening through the regions where > . The higher the modulus-mismatch, the higher and E . In other words, the larger the variation of strain between the graphene and the matrix, the lower the stiffening efficiency of graphene. Hence, AVSS could be used as a measure of stiffening efficiency. The interphase shows a stiffening mechanism that resembles a reinforcing particle as could be seen from the analysis of E and E results, in addition to enhanced load-transfer to graphene. Given its low mechanical properties compared to graphene, its low volume fraction and its modest effect on the load-transfer to the graphene, could explain its insignificant effect on the enhancement of , which is in agreement with previous reports [35,36,49].
25
However, for multi-particle RVE, it appears that there is a threshold thickness of the interphase where there is a strong interaction of the strain fields resulting in a more efficient load-transfer to the reinforcement and more enhancement of . This study suggests that even at perfect graphene and interface conditions, the stiffening will be limited to unsatisfactory levels that are relatively closer to the lower bound. The active contribution of the matrix raises questions on the durability of nanocomposites [10]. Therefore, in addition to the research efforts to enhance the quality of graphene and the interface with the matrix, additional efforts should be directed towards synthesis methods that would enhance the strain levels in the graphene. Novel full-field measurement methods at the nanoscale are required to assess strain-field distribution. Additionally, the effect of the interphase and the interaction of strain-fields needs more study for more efficient utilization of graphene as a reinforcement.
7. Conclusions The following conclusions could be withdrawn: •
AVSS is derived to express the internal state of strain in terms of the composite parameters. AVSS increases with decreasing . For the same f, AVSS is a measure of the variation in the strain levels between the matrix and the reinforcement. This could indicate the load transfer efficiency to the reinforcement.
•
The large modulus-mismatch between the graphene and the polymer matrix limits the graphene’s capability in stiffening by reducing its strain levels.
•
In the case of non-uniform strain fields, the matrix has an active contribution to stiffening through its highly strained regions. The contribution of these regions is directly proportional to the modulus-mismatch with the reinforcement.
26
•
Strain energy is a proper measure to include the effect of the non-uniform strain fields in nanocomposites. Fraction strain energy could quantitatively measure the contribution of each of the graphene and the matrix to .
•
In addition to enhancing the graphene and its interface quality, developing novel synthesis methods that would increase its strain levels is crucial in enhancing the efficiency of graphene as a reinforcement.
Acknowledgement This work was generously supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) Grant Number: NRF2016R1D1A1B03932101. A.H. is grateful to the Korea University of Technology and Education Post-doctoral fellowship.
8. References [1]
C. Lee, X. Wei, J.W. Kysar, J. Hone, Measurement of the Elastic Properties and Intrinsic Strength of Monolayer Graphene, Science. 321 (2008) 385–388. doi:10.1126/science.1157996.
[2]
M.A. Rafiee, J. Rafiee, Z. Wang, H. Song, Z.Z. Yu, N. Koratkar, Enhanced mechanical properties of nanocomposites at low graphene content, ACS Nano. 3 (2009) 3884–3890. doi:10.1021/nn9010472.
[3]
J. Liang, Y. Huang, L. Zhang, Y. Wang, Y. Ma, T. Guo, et al., Molecular-Level Dispersion of Graphene into Poly(vinyl alcohol) and Effective Reinforcement of their Nanocomposites, Adv. Funct. Mater. 19 (2009) 2297–2302. doi:10.1002/adfm.200801776.
[4]
L. Shao, J. Li, Y. Guang, Y. Zhang, H. Zhang, X. Che, et al., PVA/polyethyleneiminefunctionalized graphene composites with optimized properties, Mater. Des. 99 (2016) 27
235–242. doi:10.1016/j.matdes.2016.03.039. [5]
P. Liu, Z. Jin, G. Katsukis, L.W. Drahushuk, S. Shimizu, C.-J. Shih, et al., Layered and scrolled nanocomposites with aligned semi-infinite graphene inclusions at the platelet limit, Science. 353 (2016) 364–367. doi:10.1126/science.aaf4362.
[6]
A. Hussein, S. Sarkar, K. Lee, B. Kim, Cryogenic fracture behavior of epoxy reinforced by a novel graphene oxide/poly( p -phenylenediamine) hybrid, Compos. Part B Eng. 129 (2017) 133–142. doi:10.1016/j.compositesb.2017.07.085.
[7]
R.J. Young, I. a. Kinloch, L. Gong, K.S. Novoselov, The mechanics of graphene nanocomposites: A review, Compos. Sci. Technol. 72 (2012) 1459–1476. doi:10.1016/j.compscitech.2012.05.005.
[8]
G. Lubineau, A. Rahaman, A review of strategies for improving the degradation properties of laminated continuous-fiber/epoxy composites with carbon-based nanoreinforcements, Carbon 50 (2012) 2377–2395. doi:10.1016/j.carbon.2012.01.059.
[9]
D.G. Papageorgiou, I.A. Kinloch, R.J. Young, Mechanical Properties of Graphene and Graphene-based Nanocomposites, Prog. Mater. Sci. (2017). doi:10.1016/j.pmatsci.2017.07.004.
[10] T. Sui, N. Baimpas, I.P. Dolbnya, C. Prisacariu, A.M. Korsunsky, Multiple-lengthscale deformation analysis in a thermoplastic polyurethane, Nat. Commun. 6 (2015) 6583. doi:10.1038/ncomms7583. [11] Q.H. Zeng, A.B. Yu, G.Q. Lu, Multiscale modeling and simulation of polymer nanocomposites, Prog. Polym. Sci. 33 (2008) 191–269. doi:10.1016/j.progpolymsci.2007.09.002. [12] W.L. Azoti, A. Elmarakbi, Multiscale modelling of graphene platelets-based nanocomposite materials, Compos. Struct. 168 (2017) 313–321. doi:10.1016/j.compstruct.2017.02.022.
28
[13] K.N. Spanos, S.K. Georgantzinos, N.K. Anifantis, Mechanical properties of graphene nanocomposites: A multiscale finite element prediction, Compos. Struct. 132 (2015) 536–544. doi:10.1016/j.compstruct.2015.05.078. [14] J. Atalaya, A. Isacsson, J.M. Kinaret, Continuum Elastic Modeling of Graphene Resonators, Nano Lett. 8 (2008) 4196–4200. doi:10.1021/nl801733d. [15] O. Frank, G. Tsoukleri, J. Parthenios, K. Papagelis, I. Riaz, R. Jalil, et al., Compression Behavior of Single-Layer Graphenes, ACS Nano. 4 (2010) 3131–3138. doi:10.1021/nn100454w. [16] G. Tsoukleri, J. Parthenios, C. Galiotis, K. Papagelis, Embedded trilayer graphene flakes under tensile and compressive loading, 2D Mater. 2 (2015) 24009. doi:10.1088/2053-1583/2/2/024009. [17] G. Wang, L. Liu, Z. Dai, Q. Liu, H. Miao, Z. Zhang, Biaxial compressive behavior of embedded monolayer graphene inside flexible poly (methyl methacrylate) matrix, Carbon N. Y. 86 (2015) 69–77. doi:10.1016/j.carbon.2015.01.022. [18] C. Androulidakis, E.N. Koukaras, J. Parthenios, G. Kalosakas, K. Papagelis, C. Galiotis, Graphene flakes under controlled biaxial deformation, Sci. Rep. 5 (2016) 18219. doi:10.1038/srep18219. [19] L. Gong, I.A. Kinloch, R.J. Young, I. Riaz, R. Jalil, K.S. Novoselov, Interfacial stress transfer in a graphene monolayer nanocomposite, Adv. Mater. 22 (2010) 2694–2697. doi:10.1002/adma.200904264. [20] D.R. Paul, L.M. Robeson, Polymer nanotechnology: Nanocomposites, Polymer. 49 (2008) 3187–3204. doi:10.1016/j.polymer.2008.04.017. [21] M. Terrones, O. Martín, M. González, J. Pozuelo, B. Serrano, J.C. Cabanelas, et al., Interphases in Graphene Polymer-based Nanocomposites: Achievements and Challenges, Adv. Mater. 23 (2011) 5302–5310. doi:10.1002/adma.201102036.
29
[22] K.N. Spanos, N.K. Anifantis, Finite element prediction of stress transfer in graphene nanocomposites : The interface effect, Compos. Struct. 154 (2016) 269–276. doi:10.1016/j.compstruct.2016.07.058. [23] G. Dai, L. Mishnaevsky, Graphene reinforced nanocomposites: 3D simulation of damage and fracture, Comput. Mater. Sci. 95 (2014) 684–692. doi:10.1016/j.commatsci.2014.08.011. [24] B. Mortazavi, M. Baniassadi, J. Bardon, S. Ahzi, Modeling of two-phase random composite materials by finite element, Mori-Tanaka and strong contrast methods, Compos. Part B Eng. 45 (2013) 1117–1125. doi:10.1016/j.compositesb.2012.05.015. [25] R.D. Peng, H.W. Zhou, H.W. Wang, L. Mishnaevsky, Modeling of nano-reinforced polymer composites: Microstructure effect on Young’s modulus, Comput. Mater. Sci. 60 (2012) 19–31. doi:10.1016/j.commatsci.2012.03.010. [26] T.D. Fornes, D.R. Paul, Modeling properties of nylon 6/clay nanocomposites using composite theories, Polymer. 44 (2003) 4993–5013. doi:10.1016/S00323861(03)00471-3. [27] J. Tsai, C.T. Sun, Effect of Platelet Dispersion on the Load Transfer Efficiency in Nanoclay Composites, J. Compos. Mater. 38 (2004) 567–579. doi:10.1177/0021998304042397. [28] S. Zhang, M.L. Minus, L. Zhu, C.-P. Wong, S. Kumar, Polymer transcrystallinity induced by carbon nanotubes, Polymer. 49 (2008) 1356–1364. doi:10.1016/j.polymer.2008.01.018. [29] C.D. Wood, L. Chen, C. Burkhart, K.W. Putz, J.M. Torkelson, L.C. Brinson, Measuring interphase stiffening effects in styrene-based polymeric thin films, Polymer. 75 (2015) 161–167. doi:10.1016/j.polymer.2015.08.033. [30] F. Han, Y. Azdoud, G. Lubineau, Computational modeling of elastic properties of
30
carbon nanotube/polymer composites with interphase regions. Part I: Micro-structural characterization and geometric modeling, Comput. Mater. Sci. 81 (2014) 641–651. doi:10.1016/j.commatsci.2013.07.036. [31] S. Cheng, B. Carroll, V. Bocharova, J.-M. Carrillo, B.G. Sumpter, A.P. Sokolov, Focus: Structure and dynamics of the interfacial layer in polymer nanocomposites with attractive interactions, J. Chem. Phys. 146 (2017) 203201. doi:10.1063/1.4978504. [32] K.-H. Liao, S. Aoyama, A.A. Abdala, C. Macosko, Does Graphene Change T g of Nanocomposites?, Macromolecules. 47 (2014) 8311–8319. doi:10.1021/ma501799z. [33] C. Wan, B. Chen, Reinforcement and interphase of polymer/graphene oxide nanocomposites, J. Mater. Chem. 22 (2012) 3637. doi:10.1039/c2jm15062j. [34] M. Quaresimin, K. Schulte, M. Zappalorto, S. Chandrasekaran, Toughening mechanisms in polymer nanocomposites: From experiments to modelling, Compos. Sci. Technol. 123 (2016) 187–204. doi:10.1016/j.compscitech.2015.11.027. [35] B. Mortazavi, J. Bardon, S. Ahzi, Interphase effect on the elastic and thermal conductivity response of polymer nanocomposite materials: 3D finite element study, Comput. Mater. Sci. 69 (2013) 100–106. doi:10.1016/j.commatsci.2012.11.035. [36] N. Sheng, M.C. Boyce, D.M. Parks, G.C. Rutledge, J.I. Abes, R.E. Cohen, Multiscale micromechanical modeling of polymer/clay nanocomposites and the effective clay particle, Polymer. 45 (2004) 487–506. doi:10.1016/j.polymer.2003.10.100. [37] C.L. Tucker III, E. Liang, Stiffness predictions for unidirectional short-fiber composites: Review and evaluation, Compos. Sci. Technol. 59 (1999) 655–671. doi:10.1016/S0266-3538(98)00120-1. [38] J. Qu, M. Cherkaoui, Fundamentals of micromechanics of solids, John Wiley & Sons, 2006. [39] K. Hbaieb, Q.X. Wang, Y.H.J. Chia, B. Cotterell, Modelling stiffness of polymer/clay
31
nanocomposites, Polymer. 48 (2007) 901–909. doi:10.1016/j.polymer.2006.11.062. [40] G.P. Tandon, G.J. Weng, The effect of aspect ratio of inclusions on the elastic properties of unidirectionally aligned composites, Polym. Compos. 5 (1984) 327–333. doi:10.1002/pc.750050413. [41] C. Friebel, I. Doghri, V. Legat, General mean-field homogenization schemes for viscoelastic composites containing multiple phases of coated inclusions, Int. J. Solids Struct. 43 (2006) 2513–2541. doi:10.1016/j.ijsolstr.2005.06.035. [42] G.J. Weng, The theoretical connection between Mori-Tanaka’s theory and the HashinShtrikman-Walpole bounds, Int. J. Eng. Sci. 28 (1990) 1111–1120. doi:10.1016/00207225(90)90111-U. [43] G.P. Tandon, G.J. Weng, Average stress in the matrix and effective moduli of randomly oriented composites, Compos. Sci. Technol. 27 (1986) 111–132. doi:10.1016/0266-3538(86)90067-9. [44] M. Saad, Elasticity: Theory, Applications, and Numerics, 2nd ed., Academic Press, 2009. [45] D. Hull, T.W. Clyne, An Introduction to Composite Materials, 2nd ed., Cambridge University Press, 1996. [46] A. Almasi, M. Silani, H. Talebi, T. Rabczuk, Stochastic analysis of the interphase effects on the mechanical properties of clay/epoxy nanocomposites, Compos. Struct. 133 (2015) 1302–1312. doi:10.1016/j.compstruct.2015.07.061. [47] Digimat user’s manual, 2016. [48] H. Kim, A.A. Abdala, C.W. MacOsko, Graphene/polymer nanocomposites, Macromolecules. 43 (2010) 6515–6530. doi:10.1021/ma100572e. [49] R.S. Fertig, M.R. Garnich, Influence of constituent properties and microstructural parameters on the tensile modulus of a polymer/clay nanocomposite, Compos. Sci.
32
Technol. 64 (2004) 2577–2588. doi:10.1016/j.compscitech.2004.06.002.
33
S0263-8223(17)33319-6 https://doi.org/10.1016/j.compstruct.2018.01.023 COST 9263
To appear in:
Composite Structures
Received Date: Revised Date: Accepted Date:
11 October 2017 23 December 2017 9 January 2018
Please cite this article as: Hussein, A., Kim, B., Graphene/polymer nanocomposites: the active role of the matrix in stiffening mechanics, Composite Structures (2018), doi: https://doi.org/10.1016/j.compstruct.2018.01.023
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Graphene/polymer nanocomposites: the active role of the matrix in stiffening mechanics Abdelrahman Hussein ([email protected]) and Byungki Kim* ([email protected]) School of Mechatronics Engineering, Korea University of Technology and Education, 1600 Chungjeol-ro, Byeongcheon-myeon, Dongnam-gu, Cheonan, Chungnam, 31253 Republic of Korea. *
Corresponding author
Abstract In this study we investigate the efficiency of graphene in stiffening polymer matrix nanocomposites. The analysis focuses primarily on the effect of the matrix modulus on the strain fields, whilst assuming perfect graphene and interface conditions with no agglomeration. The matrix was found to have an active role in stiffening due to the regions with large strain levels. The upper and lower bounds were discussed from the perspective of strain distribution. The effective modulus of graphene nanocomposite was relatively closer to the lower bound, while it showed a transitional behavior towards the upper bound with increasing and the volume fraction. The matrix contribution to stiffening surpasses the graphene in terms of the strain energy due to the graphene’s low strain levels resulting from the large modulus-mismatch. A novel measure to the internal state of strain that could represent the strain variation between the graphene and the matrix was derived. Lower modulus-mismatch show lower strain variation and a better stiffening efficiency of graphene. These measures could also interpret the effect of interphase on stiffening. This study provides new insights to the analysis and design of graphene/polymer nanocomposites. Keywords: Graphene; RVE-FEM; Mori-Tanaka; Strain-energy
1
Graphene is the strongest synthetic material with tensile modulus in the order of 1 TPa and tensile strength of 130 GPa [1]. Research efforts were directed towards utilizing these impressive mechanical properties in nanocomposites [2–6]. Yet, it is reported that the reinforcement levels of nanoparticles like graphene and carbon nanotubes are “generally disappointing” [7] and might lead to the loss of structural integrity resulting in lower damage tolerance [8]. Young et al. [7] surveyed the stiffening , defined as the ratio of uniaxial composite modulus to that of the matrix / , and found that rigid matrices with ≈ 1 GPa showed ≈ 2, while more compliant ones with ≈1E-3 GPa showed ≈ 10. They discussed these results using the upper (Voigt) and lower (Reuss) bounds and attributed these results to imperfections in graphene platelets like incomplete exfoliation, agglomeration, weak interface and defects. In their very recent review [9], Young et al updated their survey and concluded that the common stiffening measure can be misleading. They deduced a direct relation between and the effective modulus of graphene, obtained by the rule of mixtures. In other words, they found that; opposite to what is common in literature that the stiffening efficiency of graphene is higher for compliant matrices, stiffer matrices show better stiffening. This is due to the utilization of the misleading measure . However, they did not provide a quantitative interpretation to this behavior from the perspective of the stiffening mechanics. Modelling and simulation is frequently used to study nanocomposites due to the sophisticated nature of deformation measurements at the nanoscale [10]. Various atomistic and multiscale models were employed in the analysis of nanocomposites [11–13]. Although free standing graphene is flexible in the out-of-plane deformation [14], the flexural rigidity of graphene embedded in the polymer matrix increases 6 orders of magnitude [15]. Further, the deformation of graphene embedded in polymer matrix is linear under relatively low tensile and compressive loading [16–18]. The reinforcing ability of graphene in polymer 2
nanocomposites was evidenced by the interfacial load-transfer following the shear-lag mechanism [19]. Thus, continuum based micromechanics framework is valid for modeling the load-transfer and the mechanics of nanocomposites [5,19–21]. In this framework, the effects of several parameters including: volume fraction, orientation, interface, aspect ratio and agglomeration were investigated [22–25]. In these studies, the effect of the matrix modulus is only discussed in terms of without considering the analysis of the microscopic strain fields assuming that the small size of the nano reinforcements will not affect the matrix [26,27]. Additionally, the matrix can interact with nanofillers at the interface resulting in confined interfacial regions or inducing transcrystallinity for semi-crystalline polymers [28–31]. These regions are called the interphase and have different thermomechanical properties than the bulk matrix [32]. It was reported that these interphase regions improve the glass transition temperature [32], modulus [33], and fracture toughness [34]. Mortazavi et al. [35] showed that the interphase has a strong stiffening effect for spherical nanoparticles, while for high aspect ratio particles like graphene and carbon nanotubes the stiffening effect is insignificant. This is in agreement with investigations on clay nanoparticles using Finite Element Modeling (FEM) [36]. However, to the best of the authors’ knowledge, the effect of interphase surrounding graphene particles on load-transfer and stiffening of polymer nanocomposites is lacking in the literature. This paper is motivated by two questions. First, what are the maximum stiffening levels that could be achieved by utilizing graphene as a reinforcement? And second, what is the role of the matrix in stiffening nanocomposites, and can it have an active contribution? We address these questions by analyzing the strain fields calculated from micromechanics based FEM. A simple formula is derived to express the strain levels within a general composite based on strain-energy. Unlike the average strain theorem, this formula could express the 3
effect of elastic properties of the constituent materials, the volume fraction of the reinforcement and the applied strain at the boundaries. The active role of the matrix is investigated in a single particle composite model then extended to 3D randomly oriented composite. The results were compared to the Voigt and Reuss bounds to investigate the efficiency of graphene in stiffening. Additionally, the effect of interphase thickness and modulus was investigated.
1. Background Indicial notation is mainly used in this article. The superscripts m, f and i are used for the matrix, reinforcement and interphase respectively. The case of no superscripts indicates the composite phase, while the case of no subscripts indicates the property along the loading 11 direction. Matrix notation is used whenever necessary. Upper-case bold-face Roman letters are used exclusively for fourth-order tensors. For composite materials, calculations were performed on a statistically representative domain of the composite microstructure called the representative volume element (RVE) with volume . The microscopic elastic-fields are non-uniform due to the perturbations of the reinforcement phase. Effective macroscopic elastic-fields are calculated through the volume averaging of the micro-fields as: =
1
(1)
Where is the volume averaged microscopic field . Unfortunately, the analytical solution of the micro-fields is a formidable task [37], therefore FEM is used for these calculations. For such a discrete system, volume averaging is calculated from: =
1
4
(2)
Where and are the field and volume of the element respectiveley. Following the direct approach, the effective stiffness tensor of the composite can be expressed as [37]: =
(3)
Where and are the volume-averaged stress and strain tensors respectively. Finding can be interpreted as finding an energetically equivalent homogenous material with strain-energy such that: 1 1 1 = = = 2 2 2
(4)
The effective stiffness tensor of the composite could be also approximated by Meanfield homogenization (MFH) methods based on semi-analytical models like the Mori-Tanaka (MT) model. However, MFH methods do not calculate the micro-fields of the RVE. For an isotropic matrix stiffness tensor reinforced by a filler with stiffness tensor
, the effective elastic properties are bounded by Hill’s bounds that could be represented in terms of the bulk and shear moduli as [38]: ≤ ≤ + ( − ) − ( − ) ≤ ≤ + ( − ) − ( − )
(5)
Where is the filler volume fraction. Nevertheless, in our analysis we will use the more common - yet coarse - approximation of the bounds in terms of Young’s modulus , since this form is physically appealing in describing the deformation mechanics associated with the bounds. The bounds can be thought of a representation of the arrangement or the configuration of the reinforcement within the matrix, that influences the distribution of the strain fields as illustrated schematically in Figure 1. The Voigt bound represents the state of
5
uniform strain where the strain in the composite is equal to that in the matrix and the reinforcement, such that = = (Figure 1a). The Voigt bound could be modeled by the rule-of-mixtures: = + (1 − )
(6)
The Reuss lower bound represents a state of uniform stress, such that = = (Figure 1c). The Reuss bound could be expressed as: 1 − $% =" + #
(7)
Nano reinforcements like graphene are arranged in short-fiber or particle configuration as shown in Figure 1b. The strain field in this configuration is non-uniform, which is the reason for the difficulty in analytical calculations. Therefore, in this study will be calculated from FEM using the direct approach Eq. (2) and (3) as well as the MT model as will be discussed in section 3.
Figure 1: Schematic representation of the main three composite configurations and their corresponding strain fields in the vicinity of the reinforcement.
2. Average strain squared (AVSS) as a measure of internal strain levels By carefully observing the particle configuration in Figure 1b, increasing the modulusmismatch between graphene and the matrix will result in reduced strain levels in the 6
graphene. As a result, the matrix regions at the edges of the graphene platelet will be overstrained to compensate for this reduction. When only macroscopic fields are considered, the average-strain theorem [38] applies, which states as: let the RVE domain be subjected to a constant far-field strain ∘ prescribed on its boundary ∂, then the volume average of the strain is constant and is equal to ∘ : = ∘
(8)
Thus, it couldn’t express the effect of the composite parameters on the internal strain levels. From the equivalence of the direct and the energy approaches (see supplementary information S1), the strain levels within a general composite could be expressed in terms of the average-strain-squared ( (AVSS) that could be approximated by: ( ( + (1 − ) = (
(9)
Eq. (9) shows that the internal strain level could be expressed in terms of AVSS, which is a function of ∘ , f and more important, the elastic properties. The lower the , the larger the strain within the composite. Furthermore, AVSS is always larger than the square of the (
average-strain except for the Voigt configuration such that: ( ≥
(
(10)
The average-strain theorem is evaluated at the boundary ∂ of the RVE through Gauss theorem. Therefore, it represents only the strain state at the boundaries, which is essentially constant. On the other side, the variation of strain between the matrix and the reinforcement is a local effect within the RVE domain. The advantages of AVSS over the average-strain theorem is that it shows the effect of different composite material parameters on the internal strain levels. As will be shown later in this article, this could be used as a measure of the efficiency of the reinforcement in stiffening. 7
3. The RVE model 3.1 The FEM model The RVE was modeled in Digimat-FE v2016.1 (Extreme Engineering, MSC Software, Belgium). The graphene was modeled as a disc [23,24,39] with an aspect ratio of ≈500. For the single particle model, the RVE is shown in Figure 2 with f of 0.387%. The FEM model was solved in ABAQUS v6.14 (Simulia, Dassault Systèmes, France). The RVE was meshed with tetragonal C3D10M elements. Since we are attempting the theoretically maximum limits of the particle configuration, the graphene was perfectly bonded to the matrix. All the models were subjected to 1.0E-3 uniaxial strain in the 11 direction. The material properties for graphene were 1.0 TPa for Young’s modulus and 0.165 for Poisson’s ratio [1]. The matrix moduli were varied between 0.25 and 2.5 GPa with Poisson’s ratio of 0.3. Additionally, we investigate the effect of aspect ratio of graphene platelet between 100 and 500 at constant f. In the FEM model, the interphase was modeled as a coating to the graphene platelets with a constant thickness. We investigated the effect of changing the interphase modulus values between 3.0 and 5.0 GPa as well as the thickness * of up to three times that of the graphene platelet, i.e. *, 2* and 3*.
Figure 2: Single-particle RVE with a cut-through showing the graphene platelet. For the 3D randomly oriented particles, we used a cubic RVE as shown in Figure 3. For this model, we used the same parameters except for f values that were in the range of 0.640.05 vol%.
8
Figure 3: Representative 3D randomly oriented particles RVE with f=0.64 % left and right the graphene distribution within. 3.2 Mean-field homogenization For the MFH calculations, the MT stiffness tensor is expressed as [38]: ,-./ = , + (, − , )0./
(11)
Where 0./ is the MT concentration tensor: 0./ = 01 [(1 − )3 + 01 ]
%$(12)
Where I is the unit tensor and 01 is the dilute concentration tensor, which could be expressed in terms of the Eshelby tensor for disc-shaped inclusion S [40] as: 01 = [3 + 5[, ]$%(, − , )]
%$(13)
The details of calculating S could be found in the supplementary material S2. For modeling the interphase we used the two-level MT homogenization scheme [41]. As shown schematically in Figure 4, first at the deepest level, the graphene (inclusion 1) is homogenized with the interphase resulting in inclusion 2, which is then homogenized with the matrix resulting in the overall composite. The deepest-level graphene/interphase homogenization to obtain the stiffness matrix ,- 6 can be calculated from: ,-6 = , +
(, − , )78 +
9
(14)
Where is the interphase volume fraction and 78 is the graphene/interphase starin concentration tensor that could be expressed as: $% 78 = ( + ) 71 6 [ 3 + 5]
(15)
And 71 6 is:
%$ $% 71 6 = [3 + 59, : (, − , )]
(16)
The highest-level homogenization is performed through: ,-./ = , + ( + )(,-6 − , )78(
(17)
With 78( the inclusion 2/matrix concentration tensor: 1 $% 78( = 71 ( [ 3 + ( + )7( ]
(18)
$% -6 $% 71 ( = [3 + 5[, ] (, − , )]
(19)
And 71 ( :
Figure 4: Illustration of the two-level MT homogenization scheme to account for the interphase as a coating to the graphene For a 3D randomly oriented RVE, the MT stiffness tensor is expressed as [42,43]: ,-./ = (1 − ), 06 + {, 0./ }
(20)
Where the braces {} stand for orientation averaging of a 4 th order tensor of 3D randomly orientated particles.
10
4. Results of single particle model 4.1 Effect of matrix modulus and volume fraction The results of for the single particle RVE are shown in Figure 5. The MT model slightly underestimates the FEM results. MT showed ≈ 1.1, while the FEM results show ≈ 1.3. It was reported that for aligned particles, MT underestimates FEM, while for 3D randomly oriented particles MT overestimates FEM [39]. That will be shown to be the case for 3D randomly oriented particles in subsequent sections. Thus, the FEM predictions are in good agreement with the MT model and the micro-fields calculated from FEM could be used for further analysis.
Figure 5: values calculated for single-particle RVE from FEM and MT model. (
The relation between and the average-strain theorem in terms of as well as AVSS is shown in Figure 6. Both values were calculated from FEM according to Eq. (2). These results (
were compared to the LHS of the approximated AVSS in Eq. (9). The results show that is constant and independent of as predicted from the average-strain theorem. On the other side, ( is not constant and is inversely proportional to as predicted from AVSS. Eq. (9) represents a good approximation of AVSS. Further, the results are also in agreement with Eq. (
(10) since ( > .
11
Figure 6: The effect of on the values of strain-squared calculated from approximate ( AVSS in Eq. (9), ( and . The main outcome of AVSS is to highlight the effect of the on the strain levels within the composite. Next, we show how this would affect the stiffening mechanics. Due to the non-uniform strain fields, the contribution of each of the graphene and the matrix to could be calculated from their respective contribution to . In the case of isotropic material, the constitutive relation between stress and strain is written as: =
? ( + @ ) 1+? 1 − 2?
(21)
Thus, Eq. (4) could be written as: 1 ? = ( + ) 2 1+? 1 − 2?
(22)
Knowing that the engineering shear strain ABC = 2BC , Eq. (22) in component form becomes [44]: 1 1 ( ? ( ( = ((B ( +C ( + D ( + (ABC + ABD + ACD )) + ( + + D )( ) 2 1+? 2 1 − 2? B C
12
(23)
Eq. (23) can be applied to FEM results in the discrete form of Eq. (2). The contribution of the matrix to could be calculated from its fraction strain energy E . Given the volume of graphene platelet F, the strain energy in the matrix can be calculated by replacing V with − F for the integration domain. Then, the fraction strain energy in the matrix E could be defined as: E =
(24)
Similarly, the fraction-strain-energy in the graphene E is defined as: E =
(25)
The relation between the fraction strain energy and in Figure 7 shows E > 0.95 and E J 0.05. Unexpectedly, E is inversely proportional to , while E is directly proportional to . In fact, the contribution of graphene can be neglected at lower values of . Therefore, the fraction strain energy and results suggest the need for a detailed analysis of the strain fields for the particle configuration.
Figure 7: The effect of on E and E . The strain field in the loading direction is shown in Figure 8 for the matrix moduli = 0.25 and 2.5 GPa. The strain field distribution is very similar in both cases; however,
13
the magnitudes are different within the graphene and the matrix regions surrounding it as can be seen in Figure 8a and b. The values of strain in the graphene platelet and the matrix regions adjacent to it are significantly lower than those regions at the platelet edges. The strain field within the graphene is shown in Figure 8c and d for = 0.25 and 2.5 GPa respectively. The magnitude of strain increases from the platelet edges to a maximum plateau at the center following the shear-lag theory [19,45]. The plateau of the strain in graphene increases with increasing [36]. According to the shear-lag theory, the load is transferred to the graphene particle through shear forces at the interface with the matrix. When decreases, the shear stresses, and thus the shear forces decrease resulting in lower pull and lower strain levels in the graphene. Figure 8e shows the volume-averaged strain in the graphene as a function of . =1.0E-3 is constant and is shown for comparison. is directly proportional to and increases from 2.44E-5 to 1.99E-4 for = 0.25 and 2.5 GPa respectively, which is orders of magnitude lower than . This explains the low E of graphene despite its very high elastic properties and its response to .
14
Figure 8: Strain fields in a section through the RVE for (a) = 0.25 GPa and (b) = 2.5 GPa, (c) and (d) the corresponding strain-fields distribution in the graphene, and (e) the and as a function of . Representative values of with respect to the position from the graphene platelet is shown in Figure 9. The regions at the edges of the graphene platelets have > , while the adjacent regions have J . Additionally, in the regions adjacent to the graphene platelet is directly proportional to , while in the edge regions is inversely proportional to . This suggests that in the edge regions is responsible for the increased AVSS with decreasing . Additionally, the varying values leads to varying stress levels as shown in Figure 9b and c. The edge regions have high stress levels, and thus, higher resistance to deformation. Therefore, contrary to what was reported in [26,27], the small size of nanoreinforcements does not prevent the stress concentration at the edge regions, and these regions are responsible for the active stiffening role of the matrix and its high E .
15
Figure 9: (a) Strain at representative positions of the matrix with respect to the graphene and their corresponding stress values at (b) = 0.25 GPa and (c) = 2.5 GPa. 4.2 Effect of aspect ratio The effect of the aspect ratio on is shown in Figure 10a. As the aspect ratio increase, increases due to the enhanced load-transfer as shown in Figure 10b. This is in agreement with the shear-lag model [26,45] and FEM reports [23,24,36]. The enhanced load-transfer could be evidenced from increasing E with the aspect ratio Figure 10c. However, E shows an inverse behavior. From the analysis of these results, it is interesting to note the similarity between the effect of the aspect ratio and the modulus-mismatch on stiffening mechanics by using the measures E and E , and the increased active role of the matrix at lower loadtransfer levels to graphene.
Figure 10: Effect of aspect ratio for a single particle RVE on (a) , (b) and (c) E and E . 16
4.3 Effect of interphase The effect of the interphase modulus and thickness t on for a single particle RVE is shown in Figure 11a. These results are compared to the case of no interphase, which is represented by = =2.5 GPa. Here we used fixed = 2.5 GPa since it showed the highest as discussed in Figure 5. Moreover, for the single particle RVE, the two-level MT homogenization underestimates the FEM results. The increase of with increasing and * is insignificant for the single particle RVE. This is due to the small filler volume fraction f [46] that results in a small interphase volume fraction = 0.794% and 2.42% for t and 3* respectively. This could be explained by analyzing and E shown in Figure 11b and d respectively. The interphase shows a maximum increase of ≈2% in at 3* and =5.0 GPa. This results in a modest increase in E , which explains the insignificant effect on . The interphase does not only affect the load-transfer to the graphene, but also affects the matrix contribution in terms of E . Figure 11c shows that, E has a value of 0.951 that reduces abruptly when introducing the interphase. In the presence of interphase, E decreases with increasing the interphase thickness. This is due to the increase of with increasing t. On the other side, it is interesting to see that E decreases, although modestly, with increasing . Further, the fraction strain energy of the interphase E (Figure 11e) increases with increasing t, while it decreases with increasing , i.e increasing the modulus mismatch with the bulk matrix. This behavior is similar to the behavior of a reinforcing phase as discussed earlier in Figure 7. Therefore, the interphase contributes to the stiffening by two mechanisms. First, it acts as a buffer zone for enhanced load-transfer to the graphene, which increases with increasing and t as can be seen by observing . Secondly, the interphase itself contributes to stiffening similar to a reinforcing phase as can be seen from E as well as the increased E with 17
increasing . This shows that the interpahse behaves as a reinforcing phase when its properties are higher than the bulk matrix, i.e. > . This discussion provides a foundation to the analysis of the effect of interpahse in the 3D RVE with multiple graphene platelets which shows more significant contribution due to the increased as will be shown in section 5.4.
Figure 11: Effect of interphase modulus and thickness t for a single particle RVE on (a) , (b) , (c) E , (d) E and E . For (a-d) the value = 2.5 GPa represents no interphase, i.e. same value to the matrix . The legend is shown in (a).
5. Results of the 3D randomly oriented particles model 5.1 Generation of 3D RVE First we point out the effect of aspect-ratio of graphene on the maximum agglomerationfree volume fraction f of 3D randomly oriented particles. Based on the seeding algorithm of Digimat-FE, the maximum possible agglomeration-free f was 0.64%. Higher volume fractions could be achived for uniformly oriented particles or in the presense of clustering
18
[39]. This suggests that in addition to physicochemical interactions, the geometry affects the agglomeration tendency of the nanoparticles where higher aspect-ratio leads to higher tendency to agglomeration. From an experimental point of view, this would affect the feasibility of using higher aspect-ratio particles. Most of the synthesis methods result in randomly oriented nanoparticles. High aspect-ratio particles will easily form clusters that would reduce their effective aspect-ratio and result in poor stiffening. Thus, there should be a compromise between the aspect-ratio and the clustering-free volume-fraction. Further, the generation of 3D randomly oriented particles would inevitebly induce partial anisotropy. By analyzing the tension applied to the three normal directions 11, 22 and 33 to obtain the three moduli E11, E22 and E33 respectively (supplementary material Figure S.3). We found a correlation between the anisotropy and the global error on orientation tensor indicator [47] (see supplementary material S.3). This indicator was found to have a direct relation on the size of the RVE and the volume fraction f. On the other side, increaseing the RVE size significantly increases the meshing difficulty and computational cost. In this study we used an RVE with edge size 3.5 times the diameter of the filler. This ensures levels of global error on orientation tensor of no more than 0.09 to ensure the highest possible isotropy. 5.2 Effect of matirx modulus and volume fraction The effect of on and is shown in Figure 12. The Voigt and Reuss bounds are plotted to show how the configuration affects the strain distribution, which in turn affects the stiffening mechanics. Figure 12a shows that the Voigt configuration has the highest since the strain in graphene is maximum as = . The Voigt configuration has the highest , which increases exponentially with decreasing as shown in Figure 12d. increases from ≈ 3 to 25 for =2.5 and 0.25 GPa respectively at f=0.64%. On the other side, the Reuss
19
configuration shows ≈ (Figure 12c). The large modulus-mismatch makes the graphene to behave like a rigid body with almost no strain, and thus, no contribution to stiffening. This can be seen from its values of ≈1 that are independent of f. The values for the particle configuration are shown in Figure 12b. The MT model slightly overestimates the FEM results for 3D randomly oriented particles [39]. The dependence of on f increases with increasing . This indicates a transition towards the Voigt behavior; i.e. increased strain levels within the filler, with increasing . As in the Voigt configuration, increases with decreasing and is dependent on f. The values of ≈2 for f=0.64%, which is in agreement with the experimental results reported in [7,48] for a modulus-mismatch of ≈1.0E3.
Figure 12: The effect of on and for different configurations: (a) and (d) Voigt, (b) and (e) particle, and (c) and (f) Reuss. Solid lines represent the MT results for the particle configuration. The legend is shown in (c). The AVSS of the RVE model with 3D randomly oriented graphene particles is shown in Figure 13. ( shows the same behavior discussed earlier in Figure 6, that is, ( is inversely (
proportional to and that ( > . However, the gap with the approximate expression of AVSS in Eq. (9) is larger. This is due to the increased effect of 3D strain state due to the 3D random orientation of graphene particles. Additionally, Figure 13 shows that ( is directly
20
proportional to f as predicted by AVSS. When f increases, the number of graphene particles increases and the number of highly strained edge regions of the matrix increases. This results in an overall increase in ( . Thus, for a given ∘ and f, higher AVSS indicates higher variation between the strain levels in the matrix and the reinforcement.
Figure 13: The effect of on AVSS calculated from Eq. (9) (solid lines), and ( and calculated from FEM for different f for the RVE 3D randomly oriented particles.
(
The fraction strain energy values for the three configurations are shown in Figure 14. E is complementary to E , i.e. E + E = 1, hence we limit the discussion to E for brevity. The values of E are shown in Figure 14d-f for completeness. For the three configurations, E is inversely proportional to f. E in the Voigt configuration (Figure 14a) is directly proportional to . This is expected since in this configuration the strain is uniform. The graphene in the Voigt configuration showed a pronounced contribution to stiffening since for only f=0.5%, E ≈ 0.03 and 0.3 for =0.25 and 2.5 respectively. On the other side, for the particle and the Reuss configurations, E is inversely proportional to . The Reuss configuration shows E ≈ 1 in Figure 14c which is in agreement with the values discussed in Figure 12c. Nevertheless, the results show its comparable behavior to the particle configuration with respect to . The particle
21
configuration in Figure 14b has E ≈ 0.99 for f=0.05% that decreases with increasing f. At f=0.64% E ≈ 0.95 and 0.83 for =0.25 and 2.5 GPa respectively. These results show the significance of considering the strain fields when analyzing the stiffening mechanics of particle configuration system like nanocomposites. Unlike the Voigt configuration, the contribution of graphene to stiffening in the particle configuration decreases with decreasing . The matrix actively contributes to stiffening due to its highly strained regions as can be seen from its surpassing E values. This active contribution increases with decreasing as the matrix is overstrained to compensate for the decreasing strain levels within the graphene. This can be also demonstrated from the AVSS results, where the increased strain levels with decreasing indicated increased variation in strain levels between the matrix and the graphene.
Figure 14: Fraction strain energy for Voigt (a) and (d), particle (b) and (e), and Reuss (c) and (f) configurations. The legend is shown in (a). 5.3 Effect of aspect ratio For 3D RVE, we investigated the effect of aspect ratio and interphase (section 5.4) for the filler volume fraction f 0.5% and 2.5 GPa. The results here are similar to those discussed
22
for single particle RVE in Figure 10. This further verifies the importance of the analysis of strain fields and the use of E and E to understand the stiffening mechanics of graphene polymer nanocomposites.
Figure 15: Effect of aspect ratio for 3D RVE on (a) , (b) and (c) E and E . 5.4 Effect of interphase For the 3D RVE, the effect of interphase is summarized in Figure 16. Here, the interpahse thickness *, 2* and 3* corresponds to interphase volume fraction fi 1.03%, 2.08% and 3.14% respectively. The effect of interphase on is shown in Figure 16a. The two-level MT homogenization scheme over estimates the FEM results, however, it shows satisfactory estimations considering the simplicity of calculation compared to FEM. It should also be noted that the effect of the interphase on in the 3D RVE is very sensitive to the global error in orientation tensor that should be minimal in order to obtain reliable results. Compared to the single particle RVE, here, the thickness shows more efficiency to enhance as can be shown from the sudden increase form * to 2*. This behavior can also be observed for , E and E KL Figure 16(b-d). However, for E , there is no abrupt increase with *. This indicates that in addition to the role of the interphase discussed in section 4.2, there is an additional mechanism for multi-particle RVE, that is, increased interaction of strain fields with
23
increasing *. The larger interaction of strain fields enhances the strain levels in the filler that results in improving the stiffness .
Figure 16: Summary of the results of 3D RVE with f=0.5% and interphase thickness t: (a) , (b) , (c) E , (d) E and (e) E . For (a-d) the value = 2.5 GPa represents no interphase, The legend is shown in (a).
6. Discussion Mean-field homogenization with Mori-Tanaka scheme is convenient for estimating the effect of microstructural parameters like modulus-mismatch, aspect-ratio and interphase on the effective elastic properties of graphene polymer nanocomposites. This is due to its generality and relative ease of calculation compared to FEM. However, to understand the stiffening and load-transfer mechanics, microscopic fields are required, and thus, FEM becomes indispensable. The motivation for this study was to understand the stiffening mechanism in graphene nanocomposites, the limits to this stiffening with respect to the upper and the lower bounds and the contribution of each of the graphene and the matrix. depends not only on the elastic properties of the graphene and the matrix, but also on their respective strain levels. This 24
dependence is appropriately expressed in terms of strain energy based measures. The configuration of the graphene affects the distribution of the strain field and subsequently affects the bounds of . The Voigt bound is the upper bound because the strain in the reinforcement is maximum and is equal to that of the matrix, whilst it is minimum for the Ruess lower bound. The particle configuration, which represents most of the experimentally prepared graphene nanocomposites, shows values close to the lower bound. This is due to the graphene’s very low E as a result of its low . The E values further decrease with increasing the modulus-mismatch with the matrix. Moreover, due to its very large aspect ratio, the maximum agglomeration free volume fraction of 3D randomly oriented particles is very limited. It appears that the main two impressive characteristics of graphene, i.e. its modulus and aspect-ratio, hinder its efficiency in stiffening nanocomposites. For the Voigt configuration, matrices with low modulus show impressive values when reinforced with graphene due to its high mechanical properties. However, in the case of nonuniform strain fields like the particle configuration is a misleading measure [20]. The matrix is overstrained to compensate for the lower due to the large modulus-mismatch. Thus, the matrix plays an active role in stiffening through the regions where > . The higher the modulus-mismatch, the higher and E . In other words, the larger the variation of strain between the graphene and the matrix, the lower the stiffening efficiency of graphene. Hence, AVSS could be used as a measure of stiffening efficiency. The interphase shows a stiffening mechanism that resembles a reinforcing particle as could be seen from the analysis of E and E results, in addition to enhanced load-transfer to graphene. Given its low mechanical properties compared to graphene, its low volume fraction and its modest effect on the load-transfer to the graphene, could explain its insignificant effect on the enhancement of , which is in agreement with previous reports [35,36,49].
25
However, for multi-particle RVE, it appears that there is a threshold thickness of the interphase where there is a strong interaction of the strain fields resulting in a more efficient load-transfer to the reinforcement and more enhancement of . This study suggests that even at perfect graphene and interface conditions, the stiffening will be limited to unsatisfactory levels that are relatively closer to the lower bound. The active contribution of the matrix raises questions on the durability of nanocomposites [10]. Therefore, in addition to the research efforts to enhance the quality of graphene and the interface with the matrix, additional efforts should be directed towards synthesis methods that would enhance the strain levels in the graphene. Novel full-field measurement methods at the nanoscale are required to assess strain-field distribution. Additionally, the effect of the interphase and the interaction of strain-fields needs more study for more efficient utilization of graphene as a reinforcement.
7. Conclusions The following conclusions could be withdrawn: •
AVSS is derived to express the internal state of strain in terms of the composite parameters. AVSS increases with decreasing . For the same f, AVSS is a measure of the variation in the strain levels between the matrix and the reinforcement. This could indicate the load transfer efficiency to the reinforcement.
•
The large modulus-mismatch between the graphene and the polymer matrix limits the graphene’s capability in stiffening by reducing its strain levels.
•
In the case of non-uniform strain fields, the matrix has an active contribution to stiffening through its highly strained regions. The contribution of these regions is directly proportional to the modulus-mismatch with the reinforcement.
26
•
Strain energy is a proper measure to include the effect of the non-uniform strain fields in nanocomposites. Fraction strain energy could quantitatively measure the contribution of each of the graphene and the matrix to .
•
In addition to enhancing the graphene and its interface quality, developing novel synthesis methods that would increase its strain levels is crucial in enhancing the efficiency of graphene as a reinforcement.
Acknowledgement This work was generously supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) Grant Number: NRF2016R1D1A1B03932101. A.H. is grateful to the Korea University of Technology and Education Post-doctoral fellowship.
8. References [1]
C. Lee, X. Wei, J.W. Kysar, J. Hone, Measurement of the Elastic Properties and Intrinsic Strength of Monolayer Graphene, Science. 321 (2008) 385–388. doi:10.1126/science.1157996.
[2]
M.A. Rafiee, J. Rafiee, Z. Wang, H. Song, Z.Z. Yu, N. Koratkar, Enhanced mechanical properties of nanocomposites at low graphene content, ACS Nano. 3 (2009) 3884–3890. doi:10.1021/nn9010472.
[3]
J. Liang, Y. Huang, L. Zhang, Y. Wang, Y. Ma, T. Guo, et al., Molecular-Level Dispersion of Graphene into Poly(vinyl alcohol) and Effective Reinforcement of their Nanocomposites, Adv. Funct. Mater. 19 (2009) 2297–2302. doi:10.1002/adfm.200801776.
[4]
L. Shao, J. Li, Y. Guang, Y. Zhang, H. Zhang, X. Che, et al., PVA/polyethyleneiminefunctionalized graphene composites with optimized properties, Mater. Des. 99 (2016) 27
235–242. doi:10.1016/j.matdes.2016.03.039. [5]
P. Liu, Z. Jin, G. Katsukis, L.W. Drahushuk, S. Shimizu, C.-J. Shih, et al., Layered and scrolled nanocomposites with aligned semi-infinite graphene inclusions at the platelet limit, Science. 353 (2016) 364–367. doi:10.1126/science.aaf4362.
[6]
A. Hussein, S. Sarkar, K. Lee, B. Kim, Cryogenic fracture behavior of epoxy reinforced by a novel graphene oxide/poly( p -phenylenediamine) hybrid, Compos. Part B Eng. 129 (2017) 133–142. doi:10.1016/j.compositesb.2017.07.085.
[7]
R.J. Young, I. a. Kinloch, L. Gong, K.S. Novoselov, The mechanics of graphene nanocomposites: A review, Compos. Sci. Technol. 72 (2012) 1459–1476. doi:10.1016/j.compscitech.2012.05.005.
[8]
G. Lubineau, A. Rahaman, A review of strategies for improving the degradation properties of laminated continuous-fiber/epoxy composites with carbon-based nanoreinforcements, Carbon 50 (2012) 2377–2395. doi:10.1016/j.carbon.2012.01.059.
[9]
D.G. Papageorgiou, I.A. Kinloch, R.J. Young, Mechanical Properties of Graphene and Graphene-based Nanocomposites, Prog. Mater. Sci. (2017). doi:10.1016/j.pmatsci.2017.07.004.
[10] T. Sui, N. Baimpas, I.P. Dolbnya, C. Prisacariu, A.M. Korsunsky, Multiple-lengthscale deformation analysis in a thermoplastic polyurethane, Nat. Commun. 6 (2015) 6583. doi:10.1038/ncomms7583. [11] Q.H. Zeng, A.B. Yu, G.Q. Lu, Multiscale modeling and simulation of polymer nanocomposites, Prog. Polym. Sci. 33 (2008) 191–269. doi:10.1016/j.progpolymsci.2007.09.002. [12] W.L. Azoti, A. Elmarakbi, Multiscale modelling of graphene platelets-based nanocomposite materials, Compos. Struct. 168 (2017) 313–321. doi:10.1016/j.compstruct.2017.02.022.
28
[13] K.N. Spanos, S.K. Georgantzinos, N.K. Anifantis, Mechanical properties of graphene nanocomposites: A multiscale finite element prediction, Compos. Struct. 132 (2015) 536–544. doi:10.1016/j.compstruct.2015.05.078. [14] J. Atalaya, A. Isacsson, J.M. Kinaret, Continuum Elastic Modeling of Graphene Resonators, Nano Lett. 8 (2008) 4196–4200. doi:10.1021/nl801733d. [15] O. Frank, G. Tsoukleri, J. Parthenios, K. Papagelis, I. Riaz, R. Jalil, et al., Compression Behavior of Single-Layer Graphenes, ACS Nano. 4 (2010) 3131–3138. doi:10.1021/nn100454w. [16] G. Tsoukleri, J. Parthenios, C. Galiotis, K. Papagelis, Embedded trilayer graphene flakes under tensile and compressive loading, 2D Mater. 2 (2015) 24009. doi:10.1088/2053-1583/2/2/024009. [17] G. Wang, L. Liu, Z. Dai, Q. Liu, H. Miao, Z. Zhang, Biaxial compressive behavior of embedded monolayer graphene inside flexible poly (methyl methacrylate) matrix, Carbon N. Y. 86 (2015) 69–77. doi:10.1016/j.carbon.2015.01.022. [18] C. Androulidakis, E.N. Koukaras, J. Parthenios, G. Kalosakas, K. Papagelis, C. Galiotis, Graphene flakes under controlled biaxial deformation, Sci. Rep. 5 (2016) 18219. doi:10.1038/srep18219. [19] L. Gong, I.A. Kinloch, R.J. Young, I. Riaz, R. Jalil, K.S. Novoselov, Interfacial stress transfer in a graphene monolayer nanocomposite, Adv. Mater. 22 (2010) 2694–2697. doi:10.1002/adma.200904264. [20] D.R. Paul, L.M. Robeson, Polymer nanotechnology: Nanocomposites, Polymer. 49 (2008) 3187–3204. doi:10.1016/j.polymer.2008.04.017. [21] M. Terrones, O. Martín, M. González, J. Pozuelo, B. Serrano, J.C. Cabanelas, et al., Interphases in Graphene Polymer-based Nanocomposites: Achievements and Challenges, Adv. Mater. 23 (2011) 5302–5310. doi:10.1002/adma.201102036.
29
[22] K.N. Spanos, N.K. Anifantis, Finite element prediction of stress transfer in graphene nanocomposites : The interface effect, Compos. Struct. 154 (2016) 269–276. doi:10.1016/j.compstruct.2016.07.058. [23] G. Dai, L. Mishnaevsky, Graphene reinforced nanocomposites: 3D simulation of damage and fracture, Comput. Mater. Sci. 95 (2014) 684–692. doi:10.1016/j.commatsci.2014.08.011. [24] B. Mortazavi, M. Baniassadi, J. Bardon, S. Ahzi, Modeling of two-phase random composite materials by finite element, Mori-Tanaka and strong contrast methods, Compos. Part B Eng. 45 (2013) 1117–1125. doi:10.1016/j.compositesb.2012.05.015. [25] R.D. Peng, H.W. Zhou, H.W. Wang, L. Mishnaevsky, Modeling of nano-reinforced polymer composites: Microstructure effect on Young’s modulus, Comput. Mater. Sci. 60 (2012) 19–31. doi:10.1016/j.commatsci.2012.03.010. [26] T.D. Fornes, D.R. Paul, Modeling properties of nylon 6/clay nanocomposites using composite theories, Polymer. 44 (2003) 4993–5013. doi:10.1016/S00323861(03)00471-3. [27] J. Tsai, C.T. Sun, Effect of Platelet Dispersion on the Load Transfer Efficiency in Nanoclay Composites, J. Compos. Mater. 38 (2004) 567–579. doi:10.1177/0021998304042397. [28] S. Zhang, M.L. Minus, L. Zhu, C.-P. Wong, S. Kumar, Polymer transcrystallinity induced by carbon nanotubes, Polymer. 49 (2008) 1356–1364. doi:10.1016/j.polymer.2008.01.018. [29] C.D. Wood, L. Chen, C. Burkhart, K.W. Putz, J.M. Torkelson, L.C. Brinson, Measuring interphase stiffening effects in styrene-based polymeric thin films, Polymer. 75 (2015) 161–167. doi:10.1016/j.polymer.2015.08.033. [30] F. Han, Y. Azdoud, G. Lubineau, Computational modeling of elastic properties of
30
carbon nanotube/polymer composites with interphase regions. Part I: Micro-structural characterization and geometric modeling, Comput. Mater. Sci. 81 (2014) 641–651. doi:10.1016/j.commatsci.2013.07.036. [31] S. Cheng, B. Carroll, V. Bocharova, J.-M. Carrillo, B.G. Sumpter, A.P. Sokolov, Focus: Structure and dynamics of the interfacial layer in polymer nanocomposites with attractive interactions, J. Chem. Phys. 146 (2017) 203201. doi:10.1063/1.4978504. [32] K.-H. Liao, S. Aoyama, A.A. Abdala, C. Macosko, Does Graphene Change T g of Nanocomposites?, Macromolecules. 47 (2014) 8311–8319. doi:10.1021/ma501799z. [33] C. Wan, B. Chen, Reinforcement and interphase of polymer/graphene oxide nanocomposites, J. Mater. Chem. 22 (2012) 3637. doi:10.1039/c2jm15062j. [34] M. Quaresimin, K. Schulte, M. Zappalorto, S. Chandrasekaran, Toughening mechanisms in polymer nanocomposites: From experiments to modelling, Compos. Sci. Technol. 123 (2016) 187–204. doi:10.1016/j.compscitech.2015.11.027. [35] B. Mortazavi, J. Bardon, S. Ahzi, Interphase effect on the elastic and thermal conductivity response of polymer nanocomposite materials: 3D finite element study, Comput. Mater. Sci. 69 (2013) 100–106. doi:10.1016/j.commatsci.2012.11.035. [36] N. Sheng, M.C. Boyce, D.M. Parks, G.C. Rutledge, J.I. Abes, R.E. Cohen, Multiscale micromechanical modeling of polymer/clay nanocomposites and the effective clay particle, Polymer. 45 (2004) 487–506. doi:10.1016/j.polymer.2003.10.100. [37] C.L. Tucker III, E. Liang, Stiffness predictions for unidirectional short-fiber composites: Review and evaluation, Compos. Sci. Technol. 59 (1999) 655–671. doi:10.1016/S0266-3538(98)00120-1. [38] J. Qu, M. Cherkaoui, Fundamentals of micromechanics of solids, John Wiley & Sons, 2006. [39] K. Hbaieb, Q.X. Wang, Y.H.J. Chia, B. Cotterell, Modelling stiffness of polymer/clay
31
nanocomposites, Polymer. 48 (2007) 901–909. doi:10.1016/j.polymer.2006.11.062. [40] G.P. Tandon, G.J. Weng, The effect of aspect ratio of inclusions on the elastic properties of unidirectionally aligned composites, Polym. Compos. 5 (1984) 327–333. doi:10.1002/pc.750050413. [41] C. Friebel, I. Doghri, V. Legat, General mean-field homogenization schemes for viscoelastic composites containing multiple phases of coated inclusions, Int. J. Solids Struct. 43 (2006) 2513–2541. doi:10.1016/j.ijsolstr.2005.06.035. [42] G.J. Weng, The theoretical connection between Mori-Tanaka’s theory and the HashinShtrikman-Walpole bounds, Int. J. Eng. Sci. 28 (1990) 1111–1120. doi:10.1016/00207225(90)90111-U. [43] G.P. Tandon, G.J. Weng, Average stress in the matrix and effective moduli of randomly oriented composites, Compos. Sci. Technol. 27 (1986) 111–132. doi:10.1016/0266-3538(86)90067-9. [44] M. Saad, Elasticity: Theory, Applications, and Numerics, 2nd ed., Academic Press, 2009. [45] D. Hull, T.W. Clyne, An Introduction to Composite Materials, 2nd ed., Cambridge University Press, 1996. [46] A. Almasi, M. Silani, H. Talebi, T. Rabczuk, Stochastic analysis of the interphase effects on the mechanical properties of clay/epoxy nanocomposites, Compos. Struct. 133 (2015) 1302–1312. doi:10.1016/j.compstruct.2015.07.061. [47] Digimat user’s manual, 2016. [48] H. Kim, A.A. Abdala, C.W. MacOsko, Graphene/polymer nanocomposites, Macromolecules. 43 (2010) 6515–6530. doi:10.1021/ma100572e. [49] R.S. Fertig, M.R. Garnich, Influence of constituent properties and microstructural parameters on the tensile modulus of a polymer/clay nanocomposite, Compos. Sci.
32
Technol. 64 (2004) 2577–2588. doi:10.1016/j.compscitech.2004.06.002.
33